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A wavelet theory for local fields and related groups
 J. GEOM. ANAL
, 2004
"... Let G be a locally compact abelian group with compact open subgroup H. The best known example of such a group is G = Qp, the field of padic rational numbers (as a group under addition), which has compact open subgroup H = Zp, the ring of padic integers. Classical wavelet theories, which require a ..."
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Cited by 25 (1 self)
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Let G be a locally compact abelian group with compact open subgroup H. The best known example of such a group is G = Qp, the field of padic rational numbers (as a group under addition), which has compact open subgroup H = Zp, the ring of padic integers. Classical wavelet theories, which require a non trivial discrete subgroup for translations, do not apply to G, which may not have such a subgroup. A wavelet theory is developed on G using coset representatives of the discrete quotient ̂G/H ⊥ to circumvent this limitation. Wavelet bases are constructed by means of an iterative method giving rise to socalled wavelet sets in the dual group ̂G. Although the Haar and Shannon wavelets are naturally antipodal in the Euclidean setting, it is observed that their analogues for G are equivalent.
Graphs, groupoids and CuntzKrieger algebras
, 1996
"... We associate to each locally finite directed graph G two locally compact groupoids G and G(?). The unit space of G is the space of onesided infinite paths in G, and G(?) is the reduction of G to the space of paths emanating from a distinguished vertex ?. We show that under certain conditions the ..."
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Cited by 25 (11 self)
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We associate to each locally finite directed graph G two locally compact groupoids G and G(?). The unit space of G is the space of onesided infinite paths in G, and G(?) is the reduction of G to the space of paths emanating from a distinguished vertex ?. We show that under certain conditions their C algebras are Morita equivalent; the groupoid C algebra C (G) is the CuntzKrieger algebra of an infinite f0; 1g matrix defined by G, and that the algebras C (G(?)) contain the C algebras used by Doplicher and Roberts in their duality theory for compact groups. We then analyse the ideal structure of these groupoid C algebras using the general theory of Renault, and calculate their Ktheory. 1 Introduction Over the past fifteen years many C algebras and classes of C algebras have been given groupoid models. Here we consider locally finite directed graphs, which may have infinitely many vertices, but only finitely many edges in and out of each vertex. We associate ...
Examples of wavelets for local fields
 Contemporary Mathematics
, 2003
"... Abstract. It is well known that the Haar and Shannon wavelets in L 2 (R) are at opposite extremes, in the sense that the Haar wavelet is localized in time but not in frequency, whereas the Shannon wavelet is localized in freqency but not in time. We present a rich setting where the Haar and Shannon ..."
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Cited by 9 (0 self)
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Abstract. It is well known that the Haar and Shannon wavelets in L 2 (R) are at opposite extremes, in the sense that the Haar wavelet is localized in time but not in frequency, whereas the Shannon wavelet is localized in freqency but not in time. We present a rich setting where the Haar and Shannon wavelets coincide and are localized both in time and in frequency. More generally, if R is replaced by a group G with certain properties, J. Benedetto and the author have proposed a theory of wavelets on G, including the construction of wavelet sets [2]. Examples of such groups G include the padic rational group G = Qp, which is simply the completion of Q with respect to a certain natural metric topology, and the Cantor dyadic group F2((t)) of formal Laurent series with coefficients 0 or 1. In this expository paper, we consider some specific examples of the wavelet theory on such groups G. In particular, we show that Shannon wavelets on G are the same as Haar wavelets on G. We also give several examples of specific groups (such as Qp and Fp((t)), for any prime number p) and of various
Appeared in Math. Ann. 257, 61–83 (1981) Fourier Series on Certain Solenoids
"... nonnegative integers, integers, rational numbers, real numbers, and complex numbers, respectively. We denote the circle group by T and will customarily parametrize this group as ..."
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nonnegative integers, integers, rational numbers, real numbers, and complex numbers, respectively. We denote the circle group by T and will customarily parametrize this group as